Recovery Homework 2: Functions
Math 281
1. Consider the sets:
• A = {Bob, Joe, Jane}
• B = {one potato, one carrot}
(a) How many different functions f : A → B exist?
(b) How many different functions g : B → A exist?
(c) How many different functions h : B → B exist?
(d) Write down all such f, g, h’s, and indicate which are injective,
surjective, bijective.
(e) Write down all pairs of f, g such that f composed with g is the
identity function on the set B. Can you observe something about
these f ’s and g’s? (i.e. are they injective, or surjective?) Can
you formulate and explain a general statement of the form:
If f ◦ g is the identity function on a set B, then f is *****
and g is *****
2. Give an example of:
(a) A function f : Z → Z.
(b) An injective function f : Z → Z.
(c) A surjective function f : Z → Z.
(d) A bijective function (different from the identity function) f : Z →
Z.
(e) A surjective but not injective function f : Z → Z.
(f) An injective but not surjective function f : Z → Z.
(g) Two functions f, g : Z → Z, neither of which is bijective, such
that their composition is your bijective function from point (2d).
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